<PRE><B>Question 149240</B>
First, we need to find the vertical asymptote(s)


&lt;b&gt; Vertical Asymptote: &lt;/b&gt;

To find the vertical asymptote, just set the denominator equal to zero and solve for x


{{{x-7=0}}} Set the denominator equal to zero



{{{x=0+7}}}Add 7 to both sides



{{{x=7}}} Combine like terms on the right side



So the vertical asymptote is {{{x=7}}}


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Now we need to find any x-intercepts



{{{y=(x-4)/(x-7)}}} Start with the given equation



{{{0=(x-4)/(x-7)}}} Plug in {{{y=0}}}



{{{0=x-4}}} Multiply both sides by {{{x-7}}}.



{{{4=x}}} Add 4 to both sides.



So the x-intercept is (4,0)



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This means that we&#39;ll have to test three regions



Region 1: 


This region is from negative infinity to the x-intercept {{{x=4}}}


So let&#39;s test the value {{{x=0}}}


{{{y=(x-4)/(x-7)}}} Start with the given equation



{{{y=(0-4)/(0-7)}}} Plug in {{{x=0}}}



{{{y=4/7}}} Simplify.

   

Since {{{4/7}}} is greater than or equal to zero, this means that &lt;font size=4&gt;&lt;b&gt;every&lt;/b&gt;&lt;/font&gt; point in the interval &lt;font size=&quot;6&quot;&gt;(&lt;/font&gt;*[Tex \LARGE -\infty,4]&lt;font size=&quot;6&quot;&gt;]&lt;/font&gt; is above the x-axis.


So the interval &lt;font size=&quot;6&quot;&gt;(&lt;/font&gt;*[Tex \LARGE -\infty,4]&lt;font size=&quot;6&quot;&gt;]&lt;/font&gt; is part of the solution to the inequality {{{(x-4)/(x-7)&gt;=0}}}


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Region 2: 


This region is from the x-intercept {{{x=4}}} to the vertical asymptote {{{x=7}}}


So let&#39;s test the value {{{x=5}}}


{{{y=(x-4)/(x-7)}}} Start with the given equation



{{{y=(5-4)/(5-7)}}} Plug in {{{x=5}}}



{{{y=-1/2}}} Simplify.

   

Since {{{-1/2}}} is &lt;font size=4&gt;&lt;b&gt;not&lt;/b&gt;&lt;/font&gt; greater than or equal to zero, this means that &lt;font size=4&gt;&lt;b&gt;every&lt;/b&gt;&lt;/font&gt; point in the interval &lt;font size=&quot;6&quot;&gt;(&lt;/font&gt;*[Tex \LARGE 4,7]&lt;font size=&quot;6&quot;&gt;)&lt;/font&gt; is below the x-axis.


So the interval &lt;font size=&quot;6&quot;&gt;(&lt;/font&gt;*[Tex \LARGE 4,7]&lt;font size=&quot;6&quot;&gt;)&lt;/font&gt; is &lt;font size=4&gt;&lt;b&gt;not&lt;/b&gt;&lt;/font&gt; part of the solution to the inequality {{{(x-4)/(x-7)&gt;=0}}}




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Region 3: 


This region is from the vertical asymptote {{{x=7}}} to positive infinity


So let&#39;s test the value {{{x=8}}}


{{{y=(x-4)/(x-7)}}} Start with the given equation



{{{y=(8-4)/(8-7)}}} Plug in {{{x=8}}}



{{{y=4}}} Simplify.

   

Since {{{4}}} is greater than or equal to zero, this means that &lt;font size=4&gt;&lt;b&gt;every&lt;/b&gt;&lt;/font&gt; point in the interval &lt;font size=&quot;6&quot;&gt;(&lt;/font&gt;*[Tex \LARGE 4,7]&lt;font size=&quot;6&quot;&gt;)&lt;/font&gt; is above the x-axis.


So the interval &lt;font size=&quot;6&quot;&gt;(&lt;/font&gt;*[Tex \LARGE 7,\infty]&lt;font size=&quot;6&quot;&gt;)&lt;/font&gt; is  part of the solution to the inequality {{{(x-4)/(x-7)&gt;=0}}}



So that means that the solution is


&lt;font size=&quot;6&quot;&gt;(&lt;/font&gt;*[Tex \LARGE -\infty,4]&lt;font size=&quot;6&quot;&gt;]&lt;/font&gt;*[Tex \LARGE \cup]&lt;font size=&quot;6&quot;&gt;(&lt;/font&gt;*[Tex \LARGE 7,\infty]&lt;font size=&quot;6&quot;&gt;)&lt;/font&gt;</PRE>